Past exam of the mathematics course of the University of Cambridge / 2016 / ib / Paper 1 / 5C / a / Solution

Put $r^2=x^2+y^2$. On the specified domain, $r>0$, and direct differentiation gives the divergence and vorticityFor example, the third component of the curl is $A(r^2-2x^2)/r^4+A(r^2-2y^2)/r^4=0$; the other two components vanish because $U$ depends only on $z$ and the transverse velocity is independent of $z$. Hence the flow is both incompressible and irrotational exactly whenThe exclusion of the axis matters: these are ordinary derivatives on the domain, with no assertion of regularity on the removed axis.